8  Applications of Cauchy’s theorem

Theorem 8.1 (Cauchy’s integral formula) Let \(f\colon U\to \C\) be a holomorphic function and assume that \(\ol D_r(z_0)\subset U\) with boundary curve \[[0,2\pi]\xrightarrow{\ga_{\6 D_r(z_0)}}\ol D_r(z_0), \ga_{\6 D_r(z_0)}(t) = z_0+re^{it}.\] Then

\[f(z) = \frac{1}{2\pi i}\int_{\6D_r(z_0)}\frac{f(\ze)}{\ze-z}d\ze,\qquad\forall z\in D_r(z_0). \tag{8.1}\]

Proof.

Covered in lectures. Check back once the chapter is concluded.

Figure 8.1: Boundary curve of largest disk fitting inside \(U\)

Theorem 8.2 (Higher Cauchy integral formula) Let \(f\colon U\to \C\) be a holomorphic function and assume that \(\ol D_r(z_0)\subset U.\) Then \(f\) is equal on \(D_r(z_0)\) to its Taylor power series

\[f(z)=\sum_{n=0}^\iy\frac{f^{(n)}(z_0)}{n!}(z-z_0)^n, \tag{8.2}\]

which has radius of convergence \(\rho\geqslant r.\) In particular, \(f\) is infinitely complex differentiable on the open set \(U\) and has a primitive on \(D_r(z_0).\) Moreover,

\[f^{(n)}(z_0)=\frac{n!}{2\pi i}\int_{\6 D_r(z_0)}\frac{f(\ze)}{(\ze-z_0)^{n+1}}d\ze. \tag{8.3}\]

Proof.

Covered in lectures. Check back once the chapter is concluded.

Theorem 8.3 (Liouville) Every bounded entire function is constant.

Proof.

Covered in lectures. Check back once the chapter is concluded.

Carl Friedrich Gauß, 1777-1855, Österreichische Nationalbibliothek. Public Domain

Theorem 8.4 (Fundamental theorem of algebra) Every non-constant polynomial \(P(z)=a_nz^n+\ldots+a_1z+a_0\) with \(a_i\in\C\) has a complex root.

Proof.

Covered in lectures. Check back once the chapter is concluded.

Definition 8.1  

1. A subset \(D\subset\C\) is path-connected if for all \(z_0, z_1\in D\) there exists a (piecewise C¹) curve \(\ga\) in \(D\) with \(\ga(0)=z_0,\) \(\ga(1)=z_1.\)
2. A path-connected subset \(D\) is simply connected if every loop in \(D\) is (freely) homotopic in \(D\) to a constant loop.

Example 8.1  

Covered in lectures. Check back once the chapter is concluded.

Theorem 8.5 Let \(f\colon U\to\C\) be a holomorphic function. Suppose that \(U\) is simply connected and let \(z_0\in U.\) Define \(F(z)\) for each \(z\in U\) by choosing a piecewise C¹ curve \(\ga\) with \(\ga(0)=z_0,\) \(\ga(1)=z\) and defining

\[F(z)=\int_\ga f(\ze)d\ze. \tag{8.4}\]

Then \(F\) is well-defined and is the unique holomorphic function on \(U\) such that \(F'=f\) and \(F(z_0)=0.\)

Proof.

Covered in lectures. Check back once the chapter is concluded.

Example 8.2  

Covered in lectures. Check back once the chapter is concluded.

Questions for further discussion

• How should \(\int_{-i}^{1+i} z^2dz\) be interpreted? What is the result?
• Why is \(\C^\t\) path-connected but not simply connected?

Hint: consider \(\frac{1}{2\pi i}\int_\ga \frac{1}{z}dz\) for a loop \(\ga\) in \(\C^\t.\) +
It is a mysterious calculus fact that \(\arctan(x)=\sum_{n=0}^\iy (-1)^n\frac{x^{2n+1}}{2n+1}\) diverges for \(|x|> 1\) although \(\arctan\colon\R\to\R\) is smooth. In terms of the principal logarithm, \(\arctan(z)=\frac{i}{2}\log\left(\frac{i+z}{i-z}\right).\) Apply this to give a geometric explanation of the divergence using Theorem 8.2.

8.1 Exercises

Exercise 8.1

Compute the following integrals:

1. \(\int_{\partial D_1(0)}\frac{\sin(z)}{z^2}dz,\)

2. \(\int_{\partial D_{1/2}(0)}\frac{\cos(z)}{z-1}dz,\)

3. \(\int_{\partial D_2(1)}\frac{\sin(\cos(z))}{z-1}dz.\)

Exercise 8.2

Let \(f\colon\C\to\C\) be entire. Suppose there is a constant \(C>0\) such that \(|f(z)|\leqslant C|z|^d\) for all \(z\in\C.\) Prove that \(f(z)\) is a polynomial of degree \(\leqslant d.\)

Hint: Generalize the proof of Liouville’s theorem.

Exercise 8.3

Let \(f\colon U\to\C\) be holomorphic and \(\ol D_r(z_0)\subset U.\) Show that \[f(z_0)=\frac{1}{2\pi}\int_0^{2\pi} f(z_0+re^{i\varphi})d\varphi\]

and interpret this equation geometrically.

Exercise 8.4

Let \(f\colon U\to\C\) be holomorphic and \(\ol D_r(z_0)\subset U.\)

1. Prove that \(g(x)=f(z_0+re^{ix})\) is a \(2\pi\)-periodic function \(\R\to\C.\)
2. Show that the Cauchy integral formula implies an absolutely and uniformly convergent Fourier expansion \[g(x)=\sum_{n=0}^\iy \ga_ne^{inx},\qquad\forall x\in\R,\] with only non-negative Fourier modes. Moreover, show that \(\ga_n=\frac{1}{2\pi}\int_0^{2\pi}g(x)e^{-inx}dx.\)$

Note. More generally, a bi-infinite Fourier series \(\sum_{n=-\iy}^{\iy}\ga_n e^{inx}\) can be obtained as a superposition \(g_1(x)+g_2(-x).\)

Exercise 8.5

Let \(f\colon U\to\C\) be holomorphic and \(\ol D_r(z_0)\subset U.\) Using polar coordinates show that for the (surface) integral over the unit disk \(\ol D_r(x_0)=\{(x,y)\in\R^2\mid (x-x_0)^2+(y-y_0)^2\leqslant r\}\) we have \[f(z_0)=\frac{1}{\pi r^2}\int_{\ol D_r(z_0)}f(x+iy)dxdy\]

and interpret this equation geometrically.

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